The shelves of your local newsagent currently offer at least one unlikely sight. A chubby, bearded man in a suit stares out from the cover of Britain's biggest-selling music magazine. His face, which looks, perhaps, a little older than his 37 years, is arranged in the kind of lugubrious expression that was once the trademark of a fellow Greater Mancunian, Les Dawson: he has the air of a man who's just been given a court summons rather than top billing.

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Singer Guy Garvey's elevation to cover-boy status is not the only improbable aspect of Elbow's late-blooming success. You could argue that in the current climate, late-blooming success itself is pretty improbable: regularly appended to Elbow's name, the phrase "the people's band" – with its weird implication that every other double-platinum artist has a fanbase largely comprised of budgerigars or golden retrievers – tells you more about journalists struggling to come to terms with the baffling concept of a band that's got big after four albums, rather than arriving in a blaze of glory, then vanishing after 18 months. More specifically, Elbow belong in the pantheon of artists who have got big as a result of one rather atypical song. That song has certain Elbow traits, being tender and suffused with melancholy: in the teeth of its adoption as a wedding anthem and default finale to the kind of ghastly TV programme in which people announce they're on "a journey", it's worth remembering One Day Like This concerns a protagonist wistfully imagining how incredible it would be to have it off once a year. But those not familiar with their earlier oeuvre should note that before the deluge, Elbow's most high-profile fan was probably John Cale, not a man you imagine spends a lot of time punching the air to uplifting stadium rock.

Too old to have their heads turned by mainstream success, but too big-hearted, maybe too grateful, to spurn it with a churlish how-do-you-like-us-now gesture, Build a Rocket Boys! sees Elbow doing perhaps the smartest thing you could under the circumstances: carrying on regardless. You could argue that Neat Little Rows represents an amalgam of the best-known moments of its predecessor, The Seldom Seen Kid – the chorus soars towards the arena's upper tiers, the guitars crunch as on Grounds for Divorce – and that Open Arms clearly has its heart set on stirring vast crowds. But it goes about it via a pleasingly serpentine route, with a lyric about a community centre – if the nation takes it to its heart, it will presumably be the first lighters-aloft anthem in history to mention both folding chairs and finger rolls – a dinky synthesised backing that recalls the sound of the combos who played on said community centre's stage, and the unexpected arrival, one minute and three seconds in, of the Hallé Youth Choir.

Tellingly, the stuff about the buffet and the seating arrangements is more emotionally affecting than the massed voices. What Elbow excel at is alighting on small details and burnishing them into things of wonder via deceptively complex arrangements (the episodic opener The Birds demonstrates how skilful the band are at incorporating the influence of their beloved prog rock, keyboard solo and all, without straying into the territory of bombast and pomp; High Ideals deftly mixes orchestration, a Bo Diddley beat and buzzing electronics) and Garvey's permanently bruised tenor. And small details abound on Build a Rocket Boys!, perhaps inevitably, given that the album is so imbued with a nostalgic love of home – if anything, it's even more deeply rooted in the north than its predecessor. Given that The Seldom Seen Kid featured a duet with Richard Hawley that used racing pigeons as a metaphor for success, that is no mean feat. Jesus Is a Rochdale Girl is an unsentimental, but beautifully drawn vignette of Garvey's pre-fame life, "with nothing to be proud of and nothing to regret", when he was possessed only of a recently broken heart and a carefully counted collection of "45 CDs". The album's emotional centre, however, is Lippy Kids, a gorgeous meditation on adolescence that recognises both the gauche awfulness of it all – the boozy aimlessness, the "simian stroll" of the teenage boy – and the fact that you may never feel quite as rawly alive again. "Do they know these days are golden?" ponders Garvey, adding: "Build a rocket, boys!"

It slips into mawkishness once, on a reprise of The Birds that sets an elderly vocalist against a cooing choral backdrop, an idea that rather overdoes the album's theme of the passing of time. For the rest of its 50 minutes, Build a Rocket Boys! makes Elbow's success seem anything but implausible. It may be that people drawn in by One Day Like This hung around because they found music that, while less straightforward, was warm and generous and inventive. If so, they'll find more of it here.